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Volume 2011 (2011), Article ID 659672, 12 pages
Examples of Stability of Tensor Products in Positive Characteristic
Universidad Autónoma, “Benito Juárez” de Oaxaca, Mexico
Received 5 June 2011; Accepted 3 July 2011
Academic Editor: V. Futorny
Copyright © 2011 A. S. Argáez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let X be projective smooth variety over an algebraically closed field k and let ℰ, ℱ be μ-semistable locally free sheaves on X. When the base field is ℂ, using transcendental methods, one can prove that the tensor product is always a μ-semistable sheaf. However, this theorem is no longer true over positive characteristic; for an analogous theorem one needs the hypothesis of strong μ-semistability; nevertheless, this hypothesis is not a necessary condition. The objective of this paper is to construct, without the strongly μ-semistability hypothesis, a family of locally free sheaves with μ-stable tensor product.
When the base field is , the Kobayashi-Hitchin correspondence ensures that a vector bundle on a complex projective variety is polystable if and only if it admits a Hermitian-Einstein metric with respect to the Kähler metric induced by (see  for curves and  for complex compact varieties). In this way, one can prove that the tensor product of Hermitian-Einstein bundles is again Hermitian-Einstein, therefore polystable, and the same is true for symmetric and exterior products. However, in positive characteristic, this is false; in , Gieseker proved the existence of stable vector bundles on curves with nonsemistable symmetric products. When these bundles are of degree zero, the nonsemistability of the symmetric product imply the nonsemistability of the tensor product. One way to solve this problem is to introduce the concept of strong -semistability. Let be a total Frobenius morphism of ; we say that is strongly -semistable if for all the pullback is -semistable with respect to the induced polarization . Under these assumptions, the tensor product of strongly -semistable bundles is again strongly -semistable (see [4, Section 7] for curves and  for general case). However, in general, there are no conditions to ensure the -semistability of a tensor product of -semistable bundles (at least not known to the author).
The aim of this work is the construction of examples, in any characteristic, of families of -stable bundles with -stable tensor products, this without the assumption of strong -stability. The key result is Proposition 3.1, which shows that, if is étale and Galois with , then is -semistable (-polystable) if and only if is -semistable (-polystable, resp.). We remark that, in positive characteristic, this result is false for an arbitrary finite morphism. Under these hypotheses, if is a line bundle on , in Corollary 3.2 we show that is -(semi)polystable if is -(semi)stable. If in addition and for all , then in Corollary 3.5 it is proved that is -stable. The other examples that are constructed come from the -invariant decomposition of . Now, the general outline of this construction is as follows. In Section 1, we observe that has a natural -invariant decomposition: where is the set of irreducible representations of over and each is a locally free sheaf of rank . Also, we have the relations where and . In Section 2, we show that is --stable for any polarization on and also that is --polystable (--semistable) whenever is --stable (--semistable, resp.). Furthermore, let be the moduli space parametrizing --stable sheaves of rank and degree , if , we proved in Theorem 3.12 that, for any irreducible representation of and , is --stable and the natural morphism given by is injective.
In Section 3, we proved that, if , then, for all , acts without fixed points on ; in particular the morphisms are étale Galois covers, thus we have nontrivial examples for Section 2. Finally, when is an abelian variety and is an isogeny of degree , Proposition 4.5 shows that, if , then each irreducible component of is an irreducible component of , where . Section 5 is devoted to prove Proposition 3.1.
1.1. Notations and Conventions
Throughout the paper, denotes an algebraically closed field and a finite group satisfying that . Also, denotes a smooth projective variety over with a fixed polarization, that is, with a fixed ample line bundle , and we denote by any divisor in the linear system .
We denote by the group algebra of with coefficients in . Also, if and are representations of over then we denote by the . We will identify vector bundles on with locally free -modules.
2. Étale Covers
Let be an étale Galois cover. Now, we recall that a locally free sheaf on is a -sheaf (see [6, page 69]) if acts on in a way compatible with the action on . Since is étale, it is flat; hence, we have that is a locally free -module with an action of . From this, we have defined a natural morphism of -algebras . Now, as we suppose by hypothesis that , Maschke's Theorem guarantees that is a semisimple -algebra of finite dimension over . We denote by the set of irreducible representations of over , that is, the set of irreducible modules, and by the set of corresponding idempotents. Thus, we have that
On the other hand, by Theorem 1(B) in [6, page 111], there is a locally free sheaf on such that , so we have natural -invariant isomorphisms ; therefore, to understand the -structure of it suffices to do it for . Thus, we have the next.
Proposition 2.1. Let be a smooth projective variety over an algebraically closed field and and étale Galois cover with Galois group . Let and define as the -module , where is the dual representation of and is the trivial representation. Then, (1) where is the set of all irreducible representations of over and . Also, one has that and ,(2) where ,(3)each is simple (i.e., ) and if ,(4)for any irreducible representation of one has that where .
Proof. We recall that as modules, where actions are given by and , respectively. Thus, we have the following invariant isomorphisms:
Now, Theorem 1 in [6, page 111] asserts that defines an equivalence between the category of locally free modules of finite rank and the category of locally free -modules of finite rank with -action. This proves that . In particular, we have that , where is the trivial -module.
On the other hand, at the generic point of , is the function field of which, by the normal basis theorem, is isomorphic to as -modules, where is the function field of . Thus, we have a natural -isomorphisms: and we conclude that is a locally free sheaf with rank .
For the last part of (1), we need to see that , but this is a consequence that the trace morphism is a invariant isomorphism.
(2) This is immediate from
(3) We observe that and from Schur's Lemma, is 0 if and 1 if (see [7, page 181]).
(4) We know that tensor functors commutes with the pull back, so we have that and, applying Theorem 1(B) in [6, page 111], we get the desired isomorphism.
Remark 2.2. Note that (4) is valid for each of the Schur functors.
3. Stable Sheaves
Let be a locally free sheaf over and its Hilbert polynomial, where is the Euler characteristic of . It is well known that this polynomial can be written as Define the degree of as and its slope by where is the rank of . Recall that a locally free sheaf is --stable (--semistable) if for all subsheaf (, resp.); also a --semistable locally free sheaf is said --polystable if it is a direct sum of --stable sheaves. Any --stable sheaf is simple, that is, , in particular, a --polystable sheaf is --stable if and only if it is a simple sheaf.
The following proposition is proved in [8, pages 62-63] under the assumption of characteristic zero in the base field, but the arguments that surround the proof are valid for any characteristic when we consider only étale covers; however, per clarity we will repeat the proof in Section 5.
Proposition 3.1. Let be a complete variety over and be an étale Galois cover, with Galois group . Let be a locally free sheaf on . Then, is --polystable if and only if is --polystable.
Corollary 3.2. Let be a line bundle on ; then; for all --(semi)stable bundle on , is --(semi)polystable bundle.
Proof. We have that the cover is étale, thus . Then, by Proposition 3.1, is --(semi)polystable because is --(semi)polystable.
Corollary 3.3. Let be a locally free --stable -module. Then, is --polystable, and is --stable if and only if for all .
Proof. We have that the cover is étale, thus , so is --polystable because is --stable. On the other hand, so is simple if and only if for all .
Now, we recall the following lemma of the theory of semistable bundles.
Lemma 3.4. Let be a --semistable sheaf; if , then is --stable.
Proof. If is not --stable, then there exists a subsheaf such that and ; thus, we have which contradicts the assumption .
Corollary 3.5. Let be integers such that, , , and suppose that is a line bundle on such that is --stable; then, for all --stable bundle on of degree and , one has that is --stable bundle.
Proof. From Proposition 3.1, we have that is --polystable with degree and rank , and from the fact that and Lemma 3.4, we have that it is --stable. Now, by Corollary 3.3, is --stable if and only if for all ; thus, for all , then is --stable.
Theorem 3.6. Let be a smooth projective variety over an algebraically closed field and let be an étale Galois cover with Galois group and the isotypical decomposition of . Then, each is --stable with respect to any ample line bundle .
Corollary 3.7. Let be a --semistable vector bundle on , then is --semistable for all . Moreover, if is --stable, then is --polystable.
Let () be the moduli space parametrizing locally free --stable sheaves (--polystables, resp.), of rank and degree zero over over .
Corollary 3.8. Let be a smooth projective variety over an algebraically closed field . Suppose that admits an étale Galois cover with Galois group . Then, for all irreducible representation of over and any polarization , one has that is non empty.
Now, on characteristic zero, we have that exterior products of the standard representation of symmetric groups are irreducible (see [9, page 31]); thus, we have the next.
Corollary 3.9. Let be a smooth projective variety over , , and let be any polarization. Suppose that the variety admits a Galois cover with Galois group, the symmetric group on d letters . Then, there exists a nonempty open set such that is --stable for all .
Proof. As the characteristic is zero, the wedge product of --stables sheaves is --polystable, in particular --semistable. So, we have defined a morphism:
given by . As the variety is an open set of , we only need to see that there is an such that is --stable.
By hypothesis, we assume the existence of an étale Galois cover with Galois group, the symmetric group on letters ; let be the standard representation of and the corresponding sheaf obtained in Theorem 3.6, and recalling that in characteristic zero the exterior algebra of the standard representation is irreducible for all , we have, from part of Proposition 2.1, that is --stable.
In particular we have the following.
Corollary 3.10. Let be a smooth projective curve of genus over a field of characteristic zero. Then, is --stable for the generic vector bundle of degree zero and .
Proof. In this case, the moduli spaces and are irreducible, so we only need to prove the existence of a Galois cover with Galois group and, by the Lefschetz Principle (see ), it suffices to do it for . Let be the fundamental group of , with base point ; on the other hand, the symmetric group is generated by one transposition and one cycle of length , so we can define a surjective morphism , for example, the one given by , , if and , , if .
Proposition 3.11. Let be projective smooth variety over an algebraically closed field and let be an étale Galois cover with Galois group , and set . Let be a --stable vector bundle. Then, the following statements are equivalent: (1) is --stable;(2) ;(3) is --stable and for .
Proof. (3.9)(2) Since is --stable, from Proposition 3.1, we have that is --polystable; on the other hand,
Thus, is a simple vector bundle if and only if for all different from the trivial representation .
(3.9)(3) Notice that is --polystable; thus, and are --polystable. Also, we have, from Proposition 3.1, that is --polystable; then is simple, and then, --stable if and only if On the other hand, we have the following relations:
Now, and, from previous relationships, we have that equality (3.9) is possible if and only if is zero if and are simple for all . This tested the statement.
Theorem 3.12. Let be integers such that and . Then, for every irreducible representation of and one has that is --stable. In addition, the natural induced morphism is injective.
Proof. From Proposition 3.1, we have that is --polystable with degree and rank , and from the fact that and Lemma 3.4, we have that it is --stable. Thus, applying Proposition 3.11, we have that is --stable for any representation .
Let us now prove the injectivity of the morphism .
Let and suppose that ; notice that this isomorphism implies that . From the --stability of , we have that these vector bundles are simple, so that . On the other hand, from Proposition 2.1, we have that and , so and we can deduce the existence of a unique representation of dimension 1 such that . From here we have that , where and, thus, and applying part (4) of Proposition 2.1, we obtain that and so is the trivial representation. However, from representation theory, the order of the cyclic group generated by the isomorphism class of should be divided by , but by hypothesis , then and is the trivial representation.
4. Examples of Varieties with Group Action: Actions in the Moduli Space of Stable Bundles
Again, let be an étale Galois cover with Galois group , thus the action of on determines an action on the moduli space () of --(semi)stables sheaves on , and from Corollary 3.3, we have a natural morphism given by , which factorizes by the quotient . The aim of this section is the study of such morphism.
Proposition 4.1. Suppose that . Then, for all integer , the natural action of on is without fixed points; in particular, for all , one has that is --stable.
Proof. Let be the subvariety of the Picard variety formed by line bundles of degree with respect to . Thus, we have defined the determinant morphism and this satisfies that , so it suffices to prove the proposition for .
Let ; from Lemma 5.2, we have that and , so Lemma 3.4 implies the --stability of ; finally, from Corollary 3.3, for all , then we conclude that there are no fixed points. The last part of the statement is consequence of Corollary 3.3.
Denote by the quotient variety and the induced morphism by . The fiber of this map is described in the following.
Corollary 4.2. Let . Then, if , then the restriction of the quotient morphism defines an isomorphism .
Proof. In order to see this, it suffices to show that which is consequence of the proof of the above proposition.
Corollary 4.3. Let be a positive integer and suppose that . Then, one has a natural injective morphism .
Consider and curves. If , then we have that and that the moduli space is smooth (see [8, Section 4.5]). So, we have the following.
Corollary 4.4. is a smooth subvariety of .
In general, there is a natural map whose image is naturally isomorphic to the quotient variety . A special case is for abelian varieties; we note that, by a theorem of Serre-Lang, every étale cover of an abelian variety has the structure of an abelian variety (see [6, page 167]).
Proposition 4.5. Let be a polarized abelian variety and an étale cyclic cover of degree . Let be an integer such that . Then, each irreducible component of is a smooth irreducible component of .
Proof. By previous Corollary 4.4, is a smooth subvariety of and by Proposition 4.1 if . Now, let ; thus, is given by ; now, by the vanishing theorem for abelian varieties in [6, page 76], we have that for all , . Then, .
5. Proof of Proposition 3.1
We will need three lemmas; on them, we will be under the assumptions of Proposition 3.1.
Let us recall an equivalent definition of the degree of an -module.
Lemma 5.1. For a locally free sheaf , In particular, if and are locally free sheaves, then one has
Proof. By Hirzebruch-Riemann-Roch formula, we have where for some ample divisor ; thus, In particular, for , we have and the first part of the lemma follows. The second part is consequence of basic properties of Chern classes.
Lemma 5.2. Let be a locally free -module. Then, and , in particular . Let be a locally free -module. Then, and , in particular .
Proof. Statements about the rank follow from the general theory of finite covers. Now, as an étale cover is affine, we have for any coherent sheaf on ; thus, and . On the other hand, by Lemma 5.1 we have that , thus . In general, for a coherent sheaf , we have that .
Finally, let be a locally free sheaf on , then .
Lemma 5.3. Let be a locally free -module. Then, is --semistable if and only if is --semistable.
Proof. If is not --semistable, then there is a submodule with , so, by Lemma 5.2, and is not --semistable.
For the converse, suppose that is not --semistable, then, by the Harder-Narasimhan filtration theorem, there exists a unique submodule such that it is --semistable and for all submodule of ; by the uniqueness, it is invariant under the action of , so, by Theorem 1 in [6, page 111], there exists an -submodule such that and, by previous lemma, .
Proof of Proposition 3.1. Suppose that is --polystable, so by the previous lemma is --semistable and by the Jordan-Holder filtration theorem there exists a destabilizing submodule such that it is --stable and , so by previous lemmas is --semistable and and so it must be a direct summand of . Let be the inclusion and projection morphisms, so taking the direct image we have -invariant morphisms with , but, now, ; hence, we have for each irreducible representation ; in particular for the trivial representation we have , then is a direct summand of .
Now, suppose that is --stable; again, by the lemma above, is --semistable, so let be a destabilizing submodulo for it. Thus, taking the sum , we obtain a -invariant --polystable submodule of which must be the pullback of a --polystable subsheaf of with the same slope, so and then .
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